Approximation of certain bivariate functions by almost Euler means of double Fourier series

R E S E A R C H

Open Access

Approximation of certain bivariate

functions by almost Euler means

of double Fourier series

Arti Rathore

*

_{and Uaday Singh}

*_{Correspondence:}

[email protected] Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

Abstract

In this paper, we study the degree of approximation of 2

π

-periodic functions of two variables, defined onT2_{= [–}

_π

_,

_π

_]×[–

_π

_,

_π

_{] and belonging to certain Lipschitz} classes, by means of almost Euler summability of their Fourier series. The degree of approximation obtained in this way depends on the modulus of continuity associated with the functions. We also derive some corollaries from our theorems.

MSC: 40C05; 41A25; 26A15

Keywords: Double Fourier series; Symmetric partial sums; Modulus of continuity; Modulus of smoothness; Lipschitz class; Zygmund class; Almost Euler means

1 Introduction

Letf(x,y) be a 2π-periodic function in each variable and Lebesgue integrable over the two-dimensional torusT2_{= [–π,π]×[–π,π]. Then the double trigonometric Fourier}

series off(x,y) is defined by

f(x,y)∼

∞

k=–∞ ∞

l=–∞

ˆ

f(k,l)ei(kx+ly), (1)

where

ˆ

f(k,l) = 1 (2π)2

π

–π π

–π

f(u,v)e–i(ku+lv)du dv

are the Fourier coefficients of the functionf.

The double sequence of symmetric rectangular partial sums associated with Fourier series off is given by

smn(x,y) = m

k=–m n

l=–n ˆ

f(k,l)ei(kx+ly),

and its integral representation is given by

smn(x,y) = 1 π2

π

–π π

–π

f(x+u,y+v)Dm(u)Dn(v)du dv, (2)

whereDk(t) =

sin(k+1₂)t

2sin(t/2) is the Dirichlet kernel.

The concept of almost convergence of sequences was introduced and studied by G.G. Lorentz in 1948 [1]. A sequence {xn} is said to be almost convergent to a limitL, if

lim

n→∞

1 n+ 1

r+n

k=r

xk=L for allr∈N.

Móricz and Rhoades [2] extended the definition of almost convergence to double se-quences of real numbers{xmn}, almost converging toL, if

lim

m,n→∞ 1 (m+ 1)(n+ 1)

q+m

k=q r+n

l=r

xkl=L for allq,r∈N.

The Euler meansEmn(x,y) of the sequence{skl(x,y)}are defined by

Emn(x,y) =

1 (1 +q1)m(1 +q2)n

m

k=0

n

l=0

m k

n l

q1m–kq2n–lskl(x,y), q1,q2> 0,

and almost Euler means of the sequence{skl(x,y)}are defined by

τ_mnrs(x,y) = 1 (1 +q1)m(1 +q2)n

m

k=0

n

l=0

m k

n l

q1m–kq2n–lSrskl(x,y),

where

Srs_kl(x,y) = 1 (k+ 1)(l+ 1)

r+k

γ=r s+l

μ=s

sγ μ(x,y).

The following function classes are well known in the literature (see [3, 4]). For 0 <α≤1, the Lipschitz classLipαis defined by

Lipα=f :T2→R|ω(f,δ) =Oδα ,

whereω(f,δ) is the modulus of continuity off, defined by

ω(f,δ) =sup

x,y

sup

(h2_+η2₎1/2_≤δ

f(x+h,y+η) –f(x,y) .

For 0 <α,β≤1, the Lipschitz classLip(α,β) is defined by

Lip(α,β) =f :T2→R|ω1,x(f,u) =O

uαandω1,y(f,v) =O

whereω1,x(f,u) andω1,y(f,v) are the partial moduli of continuity off, defined by

ω1,x(f,u) =sup

x,y

sup

|h|≤u

f(x+h,y) –f(x,y)

and

ω1,y(f,v) =sup x,y

sup

|η|≤v

f(x,y+η) –f(x,y) .

For 0 <α,β≤2, the Zygmund classZyg(α,β) is defined by

Zyg(α,β) =f:T2_→R|ω

2,x(f,u) =O

uα_andω

2,y(f,v) =O

vβ _,

whereω2,x(f,u) andω2,y(f,v) are the partial moduli of smoothness off, defined by

ω2,x(f,u) =sup

x,y

sup

|h|≤u

_{f(x+h,y) +f(x–h,y) – 2f(x,y)}

and

ω2,y(f,v) =sup

x,y

sup

|η|≤v

f(x,y+η) +f(x,y–η) – 2f(x,y) .

Here, we generalize the definitions ofLip(α,β) andZyg(α,β) given in [3] and [4], respec-tively, by introducing a new Lipschitz classLip(α,β;p) and a Zygmund classZyg(α,β;p).

LetLp_(T2_{) (p≥1) denote the spaces of Lebesgue functions on the torusT}2_{, with the}

norm defined by

f_p=

(₂1_π02π₀2π|f(x,y)|pdx dy)1p_{, p}≥_1;

ess sup_{0≤x,y≤2π}|f(x,y)|, p=∞.

Letf(x,y) be a 2π-periodic function in each variable belonging toLp(T2) (p≥1) class. Then the total integral modulus of continuity off is defined by

ωp₁(f,u,v) = sup

|h|≤u,|η|≤v

_{f(x+h,y+η) –f(x,y)}

p

while the two partial integral moduli of continuity off are defined by

ωp_1,x(f,u) =ωp₁(f,u, 0) =sup

|h|≤u

f(x+h,y) –f(x,y)_p

and

ωp_1,y(f,v) =ωp₁(f, 0,v) =sup

|η|≤v

_{f(x,y+η) –f(x,y)}

p .

The Lipschitz classLip(α,β;p) (p≥1) forα,β∈(0, 1] is defined as

We also use the notion of integral modulus of smoothness. The total integral modulus of smoothness of a functionf is defined by

ωp₂(f,u,v) = sup

|h|≤u,|η|≤v

φx,y(h,η)_p ,

where

φx,y(h,η) =f(x+h,y+η) +f(x–h,y+η) +f(x+h,y–η)

+f(x–h,y–η) – 4f(x,y).

The partial integral moduli of smoothness are defined by

ωp_2,x(f,u) =1 2ω

p

2(f,u, 0) =sup

|h|≤u

f(x+h,y) +f(x–h,y) – 2f(x,y)_p

and

ωp_2,y(f,v) =1 2ω

p

2(f, 0,v) =sup

|η|≤v

f(x,y+η) +f(x,y–η) – 2f(x,y)_p .

It is clear thatωp₂(f,u,v),ωp_2,x(f,u) andω_2,p_y(f,v) are nondecreasing functions inuandv and that

2maxω_2,p_x(f,u),ω_2,p_y(f,v) ≤ωp₂(f,u,v)≤2ωp_2,x(f,u) +ωp_2,y(f,v)

and

ωp_2,x(f,u)≤2ωp_1,x(f,u), ωp_2,y(f,v)≤2ωp_1,y(f,v). (4)

For 0 <α,β≤2, the Zygmund classZyg(α,β;p) (p≥1) is defined as

Zyg(α,β;p) :=f ∈LpT2|ωp_2,x(f,u) =Ouαandω_2,p_y(f,v) =Ovβ .

From (4) it is clear that Lip(α,β;p)⊆Zyg(α,β;p) for 0 <α,β ≤1, and similar to one-dimensional case,Lip(α,β;p) =Zyg(α,β;p) for 0 <α,β< 1, butLip(α,β;p)=Zyg(α,β;p) formax(α,β) = 1 (see, e.g., [5], p. 44).

Letω(δ) be a nondecreasing function ofδ≥0. Thenω(δ) is of the first kind if

π

δ

ω(u) u2 du=O

ω(δ) δ

, 0 <δ≤π, (5)

andω(δ) is of the second kind if

π

δ

ω(u) u2 du=O

ω(δ) δ log

π δ

, 0 <δ≤π (6)

A functionf(x,y) is said to belong to the classLip(ψ(u,v);p) (p> 1) if

f(x+u,y+v) –f(x,y)≤M

ψ(u,v) (u.v)1/p

,

where ψ(u,v) is a positive increasing function of the variablesu,vandMis a positive constant independent ofx,y,u, andv(see [6–8]).

Here, we generalize the definition ofLip(ψ(u,v);p) (p> 1) class given above by intro-ducing a new Lipschitz classLip(ψ(u,v))Lp(p> 1) defined as

f(x+u,y+v) –f(x,y)_p≤M

ψ(u,v) (u.v)1/p

. (7)

Throughout this paper we shall use the following notations:

φx,y(u,v) =

f(x+u,y+v) +f(x–u,y+v) +f(x+u,y–v)

+f(x–u,y–v) – 4f(x,y) ,

Sr_k(u) =sin((k+ 1) u

2)sin((k+ 2r+ 1)

u

2)

sin2(u/2) = r+k

γ=r

Dγ(u), (8)

Ss_l(v) =sin((l+ 1) v

2)sin((l+ 2s+ 1)

v

2)

sin2(v/2) = s+l

μ=s

Dμ(v), (9)

Rr_m(u) = m

k=0

m k

qm–k

1

(k+ 1)S r

k(u), q1> 0, (10)

Rs_n(v) = n

l=0

n l

qn–l

2

(l+ 1)S s

l(v), q2> 0. (11)

Note 1 We can easily prove thatφx,y(u,v) satisfies the following inequalities:

φx,y(u,v)≤2

ω2,x(f,u) +ω2,y(f,v)

(12)

and

φx,y(u,v)_p≤2

ωp_2,x(f,u) +ωp_2,y(f,v). (13) Móricz and Xianlianc Shi [4] studied the rate of uniform approximation of a 2π-periodic continuous function f(x,y) in the Lipschitz class Lip(α,β) and in the Zygmund class

Zyg(α,β), 0 <α,β≤1, by Cesàro meansσmnγ δof positive order of its double Fourier series. They also obtained the result for conjugate function by using the corresponding Cesàro means.

Mittal and Rhoades [3] generalized the results of [9, 10], and [4] for a 2π-periodic contin-uous functionf(x,y) in the Lipschitz classLip(α,β) and in the Zygmund classZyg(α,β), 0 <α,β≤1, by using rectangular double matrix means of its double Fourier series. Lal [11, 12] obtained results for double Fourier series using double matrix means and product matrix means.

Also, Khan [6] obtained the degree of approximation of functions belonging to the class

Lip(ψ(u,v);p) (p> 1) by Jackson type operator. Further, Khan and Ram [8] determined the degree of approximation for the functions belonging to the classLip(ψ(u,v);p) (p> 1) by means of Gauss–Weierstrass integral of the double Fourier series off(x,y). Khan et al. [7] extended the result of Khan [6] forn-dimensional Fourier series. In [13], Krasniqi deter-mined the degree of approximation of the functions belonging to the classLip(ψ(u,v);p) (p> 1) by Euler means of double Fourier series of a functionf(x,y). In fact, he generalized the result of Khan [14] for two-dimensional and forn-dimensional cases.

2 Main results

In this paper, we study the problem in more generalized function classes defined in Sect. 1 and determine the degree of approximation by almost Euler means of the double Fourier series. More precisely, we prove the following theorem.

Theorem 2.1 Let f(x,y)be a2π-periodic function in each variable belonging to Lp(T2) (1≤p<∞).Then the degree of approximation of f(x,y)by almost Euler means of its double Fourier series is given by:

(i) If bothωp_2,xandω_2,p_yare of the first kind,then τ_mnrs(x,y) –f(x,y)_p=O

ω_2,p_x

f, 1 m+ 1

+ωp_2,y

f, 1 n+ 1

.

(ii) Ifωp_2,xis of the first kind andω_2,p_yis of the second kind,then τ_mnrs(x,y) –f(x,y)_p=O

ω_2,p_x

f, 1 m+ 1

+logπ(n+ 1)ωp_2,y

f, 1 n+ 1

.

(iii) Ifωp_2,xis of the second kind andωp_2,yis of the first kind,then τ_mnrs(x,y) –f(x,y)_p=O

logπ(m+ 1)ω_2,p_x

f, 1 m+ 1

+ωp_2,y

f, 1 n+ 1

.

(iv) If bothωp_2,xandω_2,p_yare of the second kind,then τ_mnrs(x,y) –f(x,y)_p=O

logπ(m+ 1)ω_2,p_x

f, 1 m+ 1

+logπ(n+ 1)ωp_2,y

f, 1 n+ 1

.

Theorem 2.2 Let f(x,y)be a2π-periodic function in each variable belonging to L∞(T2_).

Then the degree of approximation of f(x,y)by almost Euler means of its double Fourier series is given by:

(i) If bothω2,xandω2,yare of the first kind,then

τ_mnrs(x,y) –f(x,y)_∞=O

ω2,x

f, 1 m+ 1

+ω2,y

f, 1 n+ 1

.

(ii) Ifω2,xis of the first kind andω2,yis of the second kind,then

τ_mnrs(x,y) –f(x,y)_∞=O

ω2,x

f, 1 m+ 1

+logπ(n+ 1)ω2,y

f, 1 n+ 1

.

(iii) Ifω2,xis of the second kind andω2,yis of the first kind,then

τ_mnrs(x,y) –f(x,y)_∞=O

logπ(m+ 1)ω2,x

f, 1 m+ 1

+ω2,y

f, 1 n+ 1

.

(iv) If bothω2,xandω2,yare of the second kind,then

τ_mnrs(x,y) –f(x,y)_∞=O

logπ(m+ 1)ω2,x

f, 1 m+ 1

+logπ(n+ 1)ω2,y

f, 1 n+ 1

.

Theorem 2.3 Let f(x,y)be a2π-periodic function in each variable belonging to the class

Lip(ψ(u,v))Lp(p> 1).If the positive increasing functionψ(u,v)satisfies the condition

(uv)–σ_ψ(

u,v)is nondecreasing for some1/p<σ< 1, (14)

then the degree of approximation of f(x,y)by almost Euler means of its double Fourier series is given by

τ_mnrs(x,y) –f(x,y)_p=O(m+ 1)(n+ 1)1/p

ψ

1 m+ 1,

1 n+ 1

+ (n+ 1)–σψ

1 m+ 1,π

+ (m+ 1)–σψ

π, 1 n+ 1

+(m+ 1)(n+ 1)–σ

.

Forp=∞, the classLip(ψ(u,v))Lpreduces to the classLip(ψ(u,v))_L∞, defined as

f(x+u,y+v) –f(x,y)≤Mψ(u,v).

Thus, forp=∞, we have the following theorem.

Theorem 2.4 Let f(x,y)be a2π-periodic function in each variable belonging to the class

Lip(ψ(u,v))L∞.If the positive increasing functionψ(u,v)satisfies the condition

then the degree of approximation of f(x,y)by almost Euler means of its double Fourier series is given by

τ_mnrs(x,y) –f(x,y)_p=O

ψ

1 m+ 1,

1 n+ 1

+ (n+ 1)–σψ

1 m+ 1,π

+ (m+ 1)–σ_ψ

π, 1 n+ 1

+(m+ 1)(n+ 1)–σ

.

3 Lemmas

We need the following lemmas for the proof of our theorems.

Lemma 3.1 Let Rr

m(u)and Rsn(v)be given by(10)and(11),respectively.Then

(i) Rr_m(u) =O((1 +q1)m(m+ 1))for0 <u≤_m1₊₁.

(ii) Rs

n(v) =O((1 +q2)n(n+ 1))for0 <v≤_n1₊₁.

Proof (i) For 0 <u≤_m1₊₁, usingsin(u/2)≥u/πandsinmu≤msinu, we have

Rr_m(u)=

m

k=0

m k

qm–k

1

(k+ 1)S r k(u)

=

m

k=0

m k

qm₁–k (k+ 1)

sin((k+ 1)u₂)sin((k+ 2r+ 1)u₂)

sin2(u/2)

≤ m

k=0

m k

qm₁–k (k+ 1)

(k+ 1)(k+ 2r+ 1)sin(u₂)sin(u₂)

sin2(u/2)

= m

k=0

m k

qm₁–k(k+ 2r+ 1)

= (1 +q1)m(m+ 2r+ 1)

=O(1 +q1)m(m+ 1)

. (16)

(ii) It can be proved similarly to part (i).

Lemma 3.2 Let Rr

m(u)and Rsn(v)be given by(10)and(11),respectively.Then

(i) Rr m(u) =O

(1+q1)m

(m+1)u2

for _m1₊₁<u≤π. (ii) Rs_n(v) =O

(1+q2)n

(n+1)v2

for _n+11 <v≤π.

Proof (i) For_m1₊₁<u≤π, usingsin(u/2)≥u/πandsinu≤1, we have

Rr_m(u)=

m

k=0

m k

qm₁–k (k+ 1)S

r k(u)

=

m

k=0

m k

qm–k

1

(k+ 1)

sin((k+ 1)u

2)sin((k+ 2r+ 1)

u

2)

sin2(u/2)

≤ m

k=0

m k

qm–k

1

= π

2

(m+ 1)u2

m

k=0

m+ 1 k+ 1

qm₁–k

= π

2

(m+ 1)u2

(1 +q1)m+1–qm1+1

=O

(1 +q1)m

(m+ 1)u2

. (17)

(ii) It can be proved similarly to part (i).

4 Proof of the main results

Proof of Theorem2.1 Using the integral representation ofskl(x,y) given in (2), we have

skl(x,y) –f(x,y) = 1 4π2 π 0 π 0

φx,y(u,v)Dk(u)Dl(v)du dv.

Therefore,

Srs_kl(x,y) –f(x,y) = 1 (k+ 1)(l+ 1)

r+k

γ=r s+l

μ=s

sγ μ(x,y) –f(x,y)

= 1

(k+ 1)(l+ 1) r+k

γ=r s+l

μ=s

π

0

π

0

φx,y(u,v)

4π2 Dγ(u)Dμ(v)du dv

=

π

0

π

0

φx,y(u,v)

4π2_{(k+ 1)(l+ 1)}

_r+k

γ=r Dγ(u)

_s+l

μ=s Dμ(v)

du dv.

Now

τ_mnrs(x,y) –f(x,y)

=

1 (1 +q1)m(1 +q2)n

m k=0 n l=0 m k n l

q1m–kq2n–lSrskl(x,y)

–f(x,y)

= 1 4π2 π 0 π 0

φx,y(u,v)

(1 +q1)m(1 +q2)n

_m k=0 _m k

qm–k

1 Srk(u) (k+ 1)

_n l=0 _n l

qn–l

2 Sls(v) (l+ 1)

du dv = 1 4π2 π 0 π 0

φx,y(u,v) Rr

m(u) (1 +q1)m

Rs n(v) (1 +q2)n

du dv,

which, on applying the generalized Minkowski inequality, gives

τ_mnrs(x,y) –f(x,y)_p

= 1 4π2

π

0

π

0

φx,y(u,v) Rr

m(u) (1 +q1)m

Rs n(v) (1 +q2)n

du dv p = 1 2π 2π 0 2π 0 _4π12

π

0

π

0

φx,y(u,v)

Rr m(u) (1 +q1)m

Rs n(v) (1 +q2)n

du dv p dx dy 1/p ≤ 1 4π2 π 0 π 0

φx,y(u,v)_p |

Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

= 1 4π2

1

(m+1)

0

1

(n+1)

0

+

π

1 (m+1)

1

(n+1)

0

+

1

(m+1)

0

π

1 (n+1)

+

π

1 (m+1)

π

1 (n+1)

×φx,y(u,v)_p | Rr

m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

= 1

4π2{I1+I2+I3+I4}, say. (18)

Proof of part (i): Using Lemma 3.1 and (13), we have

I1=

1

(m+1)

0

1

(n+1)

0

φx,y(u,v)_p |

Rr_m(u)| (1 +q1)m

|Rs_n(v)| (1 +q2)n

du dv

=O(m+ 1)(n+ 1) 1 (m+1)

0

1

(n+1)

0

ωp_2,x(f,u) +ωp_2,y(f,v)du dv

=O

ωp_2,x

f, 1 m+ 1

+ω_2,p_y

f, 1 n+ 1

. (19)

Using Lemma 3.1, Lemma 3.2, (5), and (13), we have

I2=

π

1 (m+1)

1

(n+1)

0

φx,y(u,v)_p |

Rr_m(u)| (1 +q1)m

|Rs_n(v)| (1 +q2)n

du dv

=O

n+ 1 m+ 1

π

1 (m+1)

1

(n+1)

0

ωp_2,x(f,u) +ωp_2,y(f,v)du dv u2

=O

n+ 1 m+ 1

π

1 (m+1)

1 n+ 1

ωp_2,x(f,u) +ωp_2,y

f, 1 n+ 1

du u2 =O 1 m+ 1

π

1 (m+1)

ωp_2,x(f,u) u2 du+ω

p

2,y

f, 1 n+ 1

π

1 (m+1)

du u2

=O

ωp_2,x

f, 1 m+ 1

+ω_2,p_y

f, 1 n+ 1

. (20)

Similarly, we have

I3=O

ωp_2,x

f, 1 m+ 1

+ω_2,p_y

f, 1 n+ 1

. (21)

Using Lemma 3.2, (5), and (13), we have

I4=

π

1 (m+1)

π

1 (n+1)

φx,y(u,v)_p |

Rr_m(u)| (1 +q1)m

|Rs_n(v)| (1 +q2)n

du dv

=O

1 (m+ 1)(n+ 1)

π

1 (m+1)

π

1 (n+1)

ωp_2,x(f,u) +ωp_2,y(f,v)du dv u2_v2

=O

1 (m+ 1)(n+ 1)

π

1 (m+1)

(n+ 1)

ωp_2,x(f,u) +ωp_2,y

f, 1 n+ 1

du u2 =O 1 m+ 1

π

1 (m+1)

ωp_2,x(f,u) u2 du+ω

p

2,y

f, 1 n+ 1

π

1 (m+1)

du u2

=O

ωp_2,x

f, 1 m+ 1

+ω_2,p_y

f, 1 n+ 1

Collecting (18)–(22), we have

τ_mnrs(x,y) –f(x,y)_p=O

ω_2,p_x

f, 1 m+ 1

+ωp_2,y

f, 1 n+ 1

,

which proves part (i).

Proof of part (ii): Using (19) and (20), we have

I1=O

ωp_2,x

f, 1 m+ 1

+ω_2,p_y

f, 1 n+ 1

, (23)

I2=O

ωp_2,x

f, 1 m+ 1

+ω_2,p_y

f, 1 n+ 1

. (24)

Using Lemma 3.1, Lemma 3.2, (6), and (13), we have

I3=

1

(m+1)

0

π

1 (n+1)

φx,y(u,v)_p |

Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

=O

m+ 1 n+ 1

1 (m+1)

0

π

1 (n+1)

ωp_2,x(f,u) +ωp_2,y(f,v)du dv v2

=O

m+ 1 n+ 1

1 (m+1)

0

(n+ 1)

ωp_2,x(f,u) +logπ(n+ 1)ωp_2,y

f, 1 n+ 1

du

=O(m+ 1)

1

(m+1)

0

ωp_2,x(f,u)du+logπ(n+ 1)ωp_2,y

f, 1 n+ 1

1 (m+1)

0

du

=O

ωp_2,x

f, 1 m+ 1

+logπ(n+ 1)ω_2,p_y

f, 1 n+ 1

. (25)

Using Lemma 3.2, (5), (6), and (13), we have

I4=

π

1 (m+1)

π

1 (n+1)

φx,y(u,v)_p |

Rr_m(u)| (1 +q1)m

|Rs_n(v)| (1 +q2)n

du dv

=O

1 (m+ 1)(n+ 1)

π

1 (m+1)

π

1 (n+1)

ωp_2,x(f,u) +ωp_2,y(f,v)du dv u2_v2

=O

1 (m+ 1)(n+ 1)

π

1 (m+1)

(n+ 1)

ωp_2,x(f,u) +logπ(n+ 1)ωp_2,y

f, 1 n+ 1

du u2 =O 1 m+ 1

π

1 (m+1)

ωp_2,x(f,u)

u2 du+log

π(n+ 1)ωp_2,y

f, 1 n+ 1

π

1 (m+1)

du u2

=O

ωp_2,x

f, 1 m+ 1

+logπ(n+ 1)ω_2,p_y

f, 1 n+ 1

. (26)

Collecting (18), (23)–(26), we have

τ_mnrs(x,y) –f(x,y)_p=O

ω_2,p_x

f, 1 m+ 1

+logπ(n+ 1)ωp_2,y

f, 1 n+ 1

,

In a similar manner, we can prove part (iii) and part (iv).

Proof of Theorem2.2 We have

τ_mnrs(x,y) –f(x,y)

= 1 4π2

π

0

π

0

φx,y(u,v)

Rr m(u) (1 +q1)m

Rs n(v) (1 +q2)n

du dv ≤ 1 4π2 π 0 π 0

φx,y(u,v) |

Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

= 1 4π2

1

(m+1)

0

1

(n+1)

0

+

π

1 (m+1)

1

(n+1)

0

+

1

(m+1)

0

π

1 (n+1)

+

π

1 (m+1)

π

1 (n+1)

×φx,y(u,v) |

Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

= 1

4π2{I1+I2+I3+I4}, say.

Using (12) and following the proof of Theorem 2.1 with supremum norm, we will get the

required result.

Proof of Theorem 2.3 Following the proof of Theorem 2.1, using the generalized Min-kowski inequality and the fact thatφx,y(u,v)∈Lip(ψ(u,v))Lp(p> 1), we have

τ_mnrs(x,y) –f(x,y)_p

= 1 4π2

π

0

π

0

φx,y(u,v) Rr

m(u) (1 +q1)m

Rs n(v) (1 +q2)n

du dv p ≤ 1 4π2 π 0 π 0

φx,y(u,v)_p |

Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

= 1 4π2

1

(m+1)

0

1

(n+1)

0

+

π

1 (m+1)

1

(n+1)

0

+

1

(m+1)

0

π

1 (n+1)

+

π

1 (m+1)

π

1 (n+1)

×Mψ(u,v) (u.v)1/p

|Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv (27)

= 1

4π2{I1+I2+I3+I4}, say. (28)

Using Lemma 3.1, we have

I1≤

1/(m+1) 0

1/(n+1) 0

Mψ(u,v) (u.v)1/p

|Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

=O(m+ 1)(n+ 1)ψ

1 m+ 1,

1 n+ 1

1/(m+1) 0

1/(n+1) 0

(u.v)–1/pdu dv

=O

ψ

1 m+ 1,

1 n+ 1

(m+ 1)(n+ 1)1/p

Using Lemma 3.1 and Lemma 3.2, we have

I2≤

1/(m+1) 0

π

1/(n+1)

Mψ(u,v) (u.v)1/p

|Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

=

1/(m+1) 0

π

1/(n+1)

M(uv)

–σ_ψ(u,v)

(u.v)1/p–σ

|Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

=O

m+ 1 n+ 1

_ψ

(_m1₊₁,π) (_m1₊₁.π)σ

1/(m+1) 0

π

1/(n+1)

(uv)σ

v2_(uv)1/pdu dv =O

ψ

1 m+ 1,π

(m+ 1)1/p(n+ 1)1/p–σ

. (30)

Similarly, we have

I3=O

ψ

π, 1 n+ 1

(m+ 1)1/p–σ(n+ 1)1/p

. (31)

Using Lemma 3.1 and Lemma 3.2, we have

I4≤

π

1/(m+1)

π

1/(n+1)

Mψ(u,v) (u.v)1/p

|Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

=

π

1/(m+1)

π

1/(n+1)

M(uv)

–σ_ψ(u,v)

(u.v)1/p–σ

|Rr m(u)| (1 +q1)m

|Rs n(v)| (1 +q2)n

du dv

=O

1 (m+ 1)(n+ 1)

ψ(π,π) (π.π)σ

π

1/(m+1)

π

1/(n+1)

(uv)σ

(uv)2+1/pdu dv

=O(m+ 1)(n+ 1)1/p–σ. (32)

Collecting (28)–(32), we have

τ_mnrs(x,y) –f(x,y)_p=O(m+ 1)(n+ 1)1/p

ψ

1 m+ 1,

1 n+ 1

+ (n+ 1)–σ_ψ

1 m+ 1,π

+ (m+ 1)–σ_ψ

π, 1 n+ 1

+(m+ 1)(n+ 1)–σ

.

Proof of Theorem2.4 We have

τ_mnrs(x,y) –f(x,y)

= 1 4π2

π

0

π

0

φx,y(u,v) Rr

m(u) (1 +q1)m

Rs n(v) (1 +q2)n

du dv ≤ 1 4π2 π 0 π 0

φx,y(u,v) |

Rr_m(u)| (1 +q1)m

|Rs_n(v)| (1 +q2)n

du dv

= 1 4π2

1

(m+1)

0

1

(n+1)

0

+

π

1 (m+1)

1

(n+1)

0

+

1

(m+1)

0

π

1 (n+1)

+

π

1 (m+1)

π

1 (n+1)

×Mψ(u,v) |R r m(u)| (1 +q1)m

|Rs_n(v)| (1 +q2)n

du dv

= 1

4π2{I1+I2+I3+I4}, say. (33)

Now we can follow the proof of Theorem 2.3 with supremum norm to get the result.

5 Corollaries

Iff ∈Zyg(α,β;p), then

ωp_2,x(f,u) =Ouα and ωp_2,y(f,v) =Ovβ. For 0 <α,β< 1,

π

δ

uα

u2 du=O

δα–1 and

π

δ

vβ

v2dv=O

δβ–1,

which implies thatuα_andvβ_{are of the first kind.}

Forα=β= 1,

π

δ

uα

u2 du=O

logπ

δ

and

π

δ

vβ

v2dv=O

logπ

δ

,

which implies thatuα_andvβ_{are of the second kind.}

Thus, Theorem 2.1 reduces to the following corollary.

Corollary 1 If f ∈Zyg(α,β;p),then

τ_mnrs(x,y) –f(x,y)_p=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

O((m+ 1)–α_{+ (n+ 1)}–β_{), 0 <α,β< 1;}

O((m+ 1)–α₊log(n+1)

n+1 ), 0 <α< 1,β= 1;

O(log_m(m₊₁+1)+ (n+ 1)–β_{), α= 1, 0 <β< 1;}

O(log_m(m₊₁+1)+log_n(₊₁n+1)), α= 1,β= 1.

Forp=∞, the Zygmund classZyg(α,β;p) reduces toZyg(α,β). In this case, from The-orem 2.2 we have the following corollary.

Corollary 2 If f ∈Zyg(α,β),then

τ_mnrs(x,y) –f(x,y)_∞=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

O((m+ 1)–α_{+ (n+ 1)}–β_{), 0 <α,β< 1;}

O((m+ 1)–α₊log(n+1)

n+1 ), 0 <α< 1,β= 1;

O(log_m(m₊₁+1)+ (n+ 1)–β_{), α= 1, 0 <β< 1;}

O(log_m(m₊₁+1)+log_n(₊₁n+1)), α= 1,β= 1.

Acknowledgements

The authors are grateful to the reviewers for their suggestions and comments for improving the manuscript of this paper.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Received: 22 September 2017 Accepted: 2 April 2018 References

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